Derivation and application of dyadic Green´s functions for planarly layered biaxial media

Derivation of the electric field dyadic Green´s function for layered anisotropic media has been a subject of interest for decades. Methods employed include state space techniques, spectral representations, eigenvector function expansions, and various combinations of these techniques. In general the solutions are complex and require the representation of an integral kernel in terms of an appropriately defined dyadic or matrix. Interface boundaries are typically aligned with Cartesian or cylindrical coordinates in order to take advantage of the known basis functions for the geometries. Initially Lee and Kong and subsequently Mudaliar and Lee presented formulations of the unbounded dyadic Green´s functions that were derived using a symmetric dyadic integral kernel that utilized the electric field vectors present in the media. Once the dyadic Green´s functions are determined, one can apply convolution to determine the electric field in the field region. The resulting triple integral can be reduced by 1) assuming a Hertzian dipole excitation and by 2) applying any one of the asymptotic analysis techniques available, such as the method of steepest descent or the stationary phase method to reduce the integral to a simple vector equation. In the present work, we develop closed form expressions for the radiation pattern of a Hertzian dipole placed above both a single isotropic-biaxial interface as well as a two layer isotropic-biaxial geometry for those points not too close to the interface. The Green´s function presented by Mudaliar and Lee is used in the two layer analysis. A dyadic Green´s function for the half space geometry is also derived. The solution and the numerical calculations ´were verified by comparing the half space transmission and reflection coefficients as well as the radiation pattern data with reflection coefficients and electric field pattern data obtained by Ergolu and King and Sandier for uniaxial and isotropic planarly layered geometries, respec- - tively. The results showed excellent agreement